Padua points and fake nodes for polynomial approximation: old, new and open problems

Stefano DE MARCHI

doi:https://doi.org/10.33205/cma.1070020

Makale detayları

Padua points and fake nodes for polynomial approximation: old, new and open problems

Yazarlar:

Stefano DE MARCHI

Yayıncı:

Tuncer ACAR

Ana Başlıklar:

Matematik

Dergi:

Constructive Mathematical Analysis 2022 Vol.5 Issue 1, pp.14-36

Belge Tipi:

Makale

Yayın Tarihi:

3/2022

Belge Dili:

İngilizce

Document ID (Org No.):

2000914116

Sıklık:

Üç Aylık

pISSN:

2651-2939

DOI:

https://doi.org/10.33205/cma.1070020

Özet:

 


Padua points, discovered in 2005 at the University of Padua, are the first set of points on the square [−1,1]2[−1,1]2 that are explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like log2(n)log2⁡(n) of the degree. 
		One of the key features of the Padua points is that they lie on a particular Lissajous curve. Other important properties of Padua points are

Unfortunately, their extension to higher dimensions is still the biggest open problem. 

The concept of mapped bases has been widely studied (cf. e.g. [35] and references therein), which turns out to be equivalent to map the interpolating nodes and then construct the approximant in the classical form without the need of resampling. The mapping technique is general, in the sense that works with any basis and can be applied to continuous, piecewise or discontinuous functions or even images.		All the proposed methods show convergence to the interpolant provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. An effective method for interpolating via mapped bases in the multivariate setting, referred as Fake Nodes Approach (FNA), has been presented in [37]. In this paper,  some interesting connection of the FNA with Padua points and families of relatives nodes, that can be used as fake nodes for multivariate approximation, are presented and we conclude with some open problems.

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Makale detayları

Padua points and fake nodes for polynomial approximation: old, new and open problems

Yazarlar: Stefano DE MARCHI

Yayıncı: Tuncer ACAR

Ana Başlıklar: Matematik

Dergi: Constructive Mathematical Analysis 2022 Vol.5 Issue 1, pp.14-36

Belge Tipi: Makale

Yayın Tarihi: 3/2022

Belge Dili: İngilizce

Document ID (Org No.): 2000914116

Sıklık: Üç Aylık

pISSN: 2651-2939

DOI: https://doi.org/10.33205/cma.1070020

Sayı

Yazarlar:

Stefano DE MARCHI

Yayıncı:

Tuncer ACAR

Ana Başlıklar:

Matematik

Dergi:

Constructive Mathematical Analysis 2022 Vol.5 Issue 1, pp.14-36

Belge Tipi:

Makale

Yayın Tarihi:

3/2022

Belge Dili:

İngilizce

Document ID (Org No.):

2000914116

Sıklık:

Üç Aylık

pISSN:

2651-2939

DOI:

https://doi.org/10.33205/cma.1070020